The nilpotence theorem for the algebraic $K$-theory of the sphere spectrum

TL;DR

This paper proves that all positive degree elements in the algebraic K-theory of the sphere spectrum are nilpotent, extending similar results to topological cyclic homology and algebraic K-theory of integers.

Contribution

It establishes the nilpotence theorem for the algebraic K-theory of the sphere spectrum and related invariants, a significant advancement in understanding their algebraic structure.

Findings

All positive degree elements in $K_{*}( ext{sphere})$ are multiplicatively nilpotent.

Similar nilpotence results hold for $TC_{*}( ext{sphere}; ext{Z}_p^{\\wedge})$ and $K_{*}( ext{Z})$.

The results reveal deep structural properties of algebraic K-theory and topological cyclic homology.

Abstract

We prove that in the graded commutative ring $K_{*}(\mathbb{S})$, all positive degree elements are multiplicatively nilpotent. The analogous statements also hold for $TC_{*}(\mathbb{S};\mathbb{Z}^{\wedge}_p)$ and $K_{*}(\mathbb{Z})$.